Probabilistic Construction of Deterntinistic Algorithnts: Approximating Packing Integer PrograDls

We consider the problem of approximating an integer program by first solving its relaxa­ tion linear program and urounding" the resulting solution. For several packing prob­ lems, we prove probabilistically that there exists an integer solution close to the optimum of the relaxation. solution. We then develop a methodology for converting such a probabilistic existence· proof to a d.eterministic approximation algQ:rithm. The methodology mimics the existence proof in a very strong sense. o. Motivations and Main. Res,ults Some of the earliest effo,rts in integer pro­ gramming involved solv'ing the underlying relaxation linear program, and using the solu­ tion to try to find the integer optimum. In general, this does not work well [14]. Recently, Aharoni & ale [1] studied the rela­ tions between the optimum of an inte,g;er pra­ gram and that of its relaxation" for a class of hypergraph matching and covering problems. We consider several packing integer programs arising in combinatorial optimization and the design of integrated circuits. In each case we compare the integer optimum to the relaxa­ tion optimum, and use this information to develop an approximation algorithm for the integer program.